Finding Speed of Stream (Pair of Linear Equations Method)

Video Explanation

Question

A person rows at 5 km/h in still water. He takes three times as long to go 40 km upstream as to go 40 km downstream. Find the speed of the stream.

Solution

Step 1: Concept

Time = Distance / Speed

Step 2: Let Variables

Let speed of stream = \(x\) km/h

Upstream speed = \(5 – x\), Downstream speed = \(5 + x\)

Step 3: Form Equation

\[ \frac{40}{5 – x} = 3 \cdot \frac{40}{5 + x} \]

Cancel 40:

\[ \frac{1}{5 – x} = \frac{3}{5 + x} \]

Step 4: Convert into Linear Form

Let:

\[ a = \frac{1}{5 – x}, \quad b = \frac{1}{5 + x} \]

Then:

\[ a = 3b \quad (1) \]

Also,

\[ \frac{1}{a} = 5 – x,\quad \frac{1}{b} = 5 + x \]

Add:

\[ \frac{1}{a} + \frac{1}{b} = 10 \]

Multiply by \(ab\):

\[ a + b = 10ab \quad (2) \]

Step 5: Solve Linear Equations

From (1):

\[ a = 3b \]

Substitute in (2):

\[ 3b + b = 10(3b \cdot b) \]

\[ 4b = 30b^2 \]

\[ 30b^2 – 4b = 0 \]

\[ b(30b – 4) = 0 \]

\[ b = \frac{4}{30} = \frac{2}{15} \]

a = 3b = \frac{6}{15} = \frac{2}{5}

Step 6: Back Substitute

\[ 5 + x = \frac{1}{b} = \frac{15}{2} \]

\[ x = \frac{15}{2} – 5 = \frac{5}{2} \]

Conclusion

\[ \text{Speed of stream} = 2.5 \text{ km/h} \]

Verification

Upstream speed = 2.5 km/h → time = 16 hrs

Downstream speed = 7.5 km/h → time ≈ 5.33 hrs

Ratio ≈ 3 : 1 ✔